Basic Calculus SHS Q3 Lesson 6 - Special Limits PDF

Basic Calculus Capstone Project Science, Science, Technology, Technology, Engineering, Engineering, and Mathematics and Mathematics People have different ways of examining the genuineness of a pearl, which is culture...

Basic Calculus Capstone Project Science, Science, Technology, Technology, Engineering, Engineering, and Mathematics and Mathematics People have different ways of examining the genuineness of a pearl, which is cultured from the soft mantle of a mollusk, like clams. 2 Some may closely observe its surface using a magnifier. Some may feel it by rubbing it in their hands. Some may drop it and observe how it bounces. 3  In solving the limit of a function whose function value by direct substitution takes 𝟎 the indeterminate form of type , what 𝟎 method do you already know? 4 Similar to fake pearls, we may be deceived by the value of the function by direct substitution. Let us discover other methods in finding the limit of these type of functions. 5 How do we evaluate a special limit? 6  Learning Competency At the end of the lesson, you should be able to do the following: 𝐬𝐢𝐧 𝒕 Evaluate limits involving the expressions , 𝒕 𝟏−𝐜𝐨𝐬 𝒕 𝒆𝒕 −𝟏 and, using table of values 𝒕 𝒕 (STEM_BC11LC-IIIb-2). 7  Learning Objectives At the end of the lesson, you should be able to do the following: sin 𝑥 1−cos 𝑥 Illustrate the limits of 𝑓 𝑥 = , 𝑔 𝑥 = , 𝑥 𝑥 𝑒 𝑥 −1 and ℎ 𝑥 = as 𝑥 approaches 0 using a table of 𝑥 values or the graph of the function. Evaluate limits of functions involving the sin 𝑡 1−cos 𝑡 𝑒 𝑡 −1 expressions , , and. 𝑡 𝑡 𝑡 8  𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝒕 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒙 𝒕→𝟎 𝒕 sin 𝑥 The graph of 𝑓 𝑥 = is shown below. What can you say 𝑥 about the graph? What is its limit value as 𝑥 approaches 0? 9  𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝒕 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒙 𝒕→𝟎 𝒕 sin 𝑥 lim =1 𝑥→0 𝑥 10  𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝒕 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒙 𝒕→𝟎 𝒕 Let us have another example that shows that the limit of sin 𝑡 𝑓 𝑥 = as 𝑥 approaches 𝑐 is 1, provided that 𝒕 is a 𝑡 𝟎 function of 𝒙 and 𝒇(𝒄) is indeterminate of type. 𝟎 11  𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝒕 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒙 𝒕→𝟎 𝒕 sin(𝑥+1) Consider the function 𝑓 𝑥 = and let us evaluate 𝑥+1 lim 𝑓(𝑥) using table of values. 𝑥→−1 left side of −𝟏 right side of −𝟏 𝑥 𝑓(𝑥) 𝑥 𝑓(𝑥) −1.5 0.9588510772 −0.5 0.9588510772 −1.1 0.9983341665 −0.9 0.9983341665 −1.01 0.9999833334 −0.99 0.9999833334 −1.001 0.9999998333 −0.999 0.9999998333 −1.0001 0.9999999983 −0.9999 0.9999999983 12  𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝒕 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒙 𝒕→𝟎 𝒕 left side of −𝟏 right side of −𝟏 𝑥 𝑓(𝑥) 𝑥 𝑓(𝑥) −1.5 0.9588510772 −0.5 0.9588510772 −1.1 0.9983341665 −0.9 0.9983341665 −1.01 0.9999833334 −0.99 0.9999833334 −1.001 0.9999998333 −0.999 0.9999998333 −1.0001 0.9999999983 −0.9999 0.9999999983 sin(𝑥+1) Based on the tables, we have lim = 1. 𝑥→−1 𝑥+1 13  𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝒕 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒙 𝒕→𝟎 𝒕 We have the following special limit. 𝐬𝐢𝐧 𝒕 𝐥𝐢𝐦 = 𝟏. 𝒕→𝟎 𝒕 Also, 𝒕 𝐥𝐢𝐦 =𝟏 𝒕→𝟎 𝐬𝐢𝐧 𝒕 𝒕 𝟏 𝐥𝐢𝐦 𝟏 𝟏 since 𝐥𝐢𝐦 = 𝐥𝐢𝐦 𝐬𝐢𝐧 𝒕 = 𝒕→𝟎 𝐬𝐢𝐧 𝒕 = = 𝟏. 𝒕→𝟎 𝐬𝐢𝐧 𝒕 𝒕→𝟎 𝐥𝐢𝐦 𝒕 𝟏 𝒕 𝒕→𝟎 14  𝟏−𝐜𝐨𝐬 𝒙 𝟏−𝐜𝐨𝐬 𝒕 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒙 𝒙→𝒄 𝒕 1−cos 𝑥 The graph of 𝑓 𝑥 = is shown below. What can you 𝑥 say about the graph? What is its limit value as 𝑥 approaches 0? 15  𝟏−𝐜𝐨𝐬 𝒙 𝟏−𝐜𝐨𝐬 𝒕 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒙 𝒙→𝒄 𝒕 1 − cos 𝑥 lim =0 𝑥→0 𝑥 16  𝟏−𝐜𝐨𝐬 𝒙 𝟏−𝐜𝐨𝐬 𝒕 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒙 𝒙→𝒄 𝒕 Let us have another example showing that the limit of 1−cos 𝑡 𝑓 𝑥 = as 𝑥 approaches 𝑐 is 1, provided that 𝒕 is a 𝑡 𝟎 function of 𝒙 and 𝒇(𝒄) is indeterminate of type. 𝟎 17  𝟏−𝐜𝐨𝐬 𝒙 𝟏−𝐜𝐨𝐬 𝒕 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒙 𝒙→𝒄 𝒕 1−cos(𝑥+1) Consider the function 𝑓 𝑥 = and let us evaluate 𝑥+1 lim 𝑓(𝑥) using table of values. 𝑥→−1 left side of −𝟏 right side of −𝟏 𝑥 𝑓(𝑥) 𝑥 𝑓(𝑥) −1.5 −0.2448348762 −0.5 0.2448348762 −1.1 −0.0499583472 −0.9 0.0499583472 −1.01 −0.0049999583 −0.99 0.0049999583 −1.001 −0.0004999999 −0.999 0.0004999999 −1.0001 −0.00005 −0.9999 0.00005 18  𝟏−𝐜𝐨𝐬 𝒙 𝟏−𝐜𝐨𝐬 𝒕 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒙 𝒙→𝒄 𝒕 left side of −𝟏 right side of −𝟏 𝑥 𝑓(𝑥) 𝑥 𝑓(𝑥) −1.5 −0.2448348762 −0.5 0.2448348762 −1.1 −0.0499583472 −0.9 0.0499583472 −1.01 −0.0049999583 −0.99 0.0049999583 −1.001 −0.0004999999 −0.999 0.0004999999 −1.0001 −0.00005 −0.9999 0.00005 1−cos(𝑥+1) Based on the tables, we have lim = 0. 𝑥→−1 𝑥+1 19  𝟏−𝐜𝐨𝐬 𝒙 𝟏−𝐜𝐨𝐬 𝒕 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒙 𝒙→𝒄 𝒕 We have the special limit 𝟏−𝐜𝐨𝐬 𝒕 𝐥𝐢𝐦 = 𝟎. 𝒕→𝟎 𝒕 20  𝒆𝒙 −𝟏 𝒆𝒕 −𝟏 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒙 𝒕→𝒄 𝒕 𝑒 𝑥 −1 The graph of 𝑓 𝑥 = is shown below. What can you 𝑥 say about the graph? What is its limit value as 𝑥 approaches 0? 21  𝒆𝒙 −𝟏 𝒆𝒕 −𝟏 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒕 𝒙 𝒕→𝒄 𝑒𝑥 − 1 lim =1 𝑥→0 𝑥 22  𝒆𝒙 −𝟏 𝒆𝒕 −𝟏 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒕 𝒙 𝒕→𝒄 Let us have another example showing that the limit of 𝑒 𝑡 −1 𝑓 𝑥 = as 𝑥 approaches 𝑐 is 1, provided that 𝒕 is a 𝑡 𝟎 function of 𝒙 and 𝒇(𝒄) is indeterminate of type. 𝟎 23  𝒆𝒙 −𝟏 𝒆𝒕 −𝟏 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒕 𝒙 𝒕→𝒄 𝑒 𝑥+1 −1 Consider the function 𝑓 𝑥 =. Let us evaluate 𝑥+1 lim 𝑓(𝑥) using table of values. 𝑥→−1 left side of −𝟏 right side of −𝟏 𝑥 𝑓(𝑥) 𝑥 𝑓(𝑥) −1.5 0.7869386806 −0.5 1.2974425414 −1.1 0.9516258196 −0.9 1.0517091808 −1.01 0.9950166251 −0.99 1.0050167084 −1.001 0.9995001666 −0.999 1.0005001667 −1.0001 0.9999500017 −0.9999 1.0000500017 24  𝒆𝒙 −𝟏 𝒆𝒕 −𝟏 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒕 𝒙 𝒕→𝒄 left side of −𝟏 right side of −𝟏 𝑥 𝑓(𝑥) 𝑥 𝑓(𝑥) −1.5 0.7869386806 −0.5 1.2974425414 −1.1 0.9516258196 −0.9 1.0517091808 −1.01 0.9950166251 −0.99 1.0050167084 −1.001 0.9995001666 −0.999 1.0005001667 −1.0001 0.9999500017 −0.9999 1.0000500017 𝑒 𝑥+1 −1 Based on the table, we have lim = 1. 𝑥→−1 𝑥+1 25  𝒆𝒙 −𝟏 𝒆𝒕 −𝟏 The Function 𝒇 𝒙 = and the Value of 𝐥𝐢𝐦 𝒕 𝒙 𝒕→𝒄 We have the special limit 𝒆𝒕 −𝟏 𝐥𝐢𝐦 = 𝟏. 𝒕→𝟎 𝒕 Also, 𝒕 𝐥𝐢𝐦 𝒕 =𝟏 𝒕→𝟎 𝒆 −𝟏 𝒕 𝟏 𝐥𝐢𝐦 𝟏 𝟏 since 𝐥𝐢𝐦 𝒕 = 𝐥𝐢𝐦 𝒆𝒕−𝟏 = 𝒕→𝟎 𝒆𝒕 −𝟏 = = 𝟏. 𝒕→𝟎 𝒆 −𝟏 𝒕→𝟎 𝐥𝐢𝐦 𝟏 𝒕 𝒕→𝟎 𝒕 26 How do we evaluate a special limit without the use of table of values or graphs? 27  𝐬𝐢𝐧 𝒕 Steps in Evaluating the Limits of Functions Involving the Expressions , 𝒕 𝟏−𝐜𝐨𝐬 𝒕 𝒆𝒕−𝟏 , and 𝒕 𝒕 0 1. Check if the function value is indeterminate of type at 0 𝑥 = 𝑐. 28  𝐬𝐢𝐧 𝒕 Steps in Evaluating the Limits of Functions Involving the Expressions , 𝒕 𝟏−𝐜𝐨𝐬 𝒕 𝒆𝒕−𝟏 , and 𝒕 𝒕 sin 𝑥 2. Manipulate the function to obtain the expressions , 𝑥 1−cos 𝑥 𝑒 𝑥 −1 , and. This could be done by using the 𝑥 𝑥 following techniques. 29  𝐬𝐢𝐧 𝒕 Steps in Evaluating the Limits of Functions Involving the Expressions , 𝒕 𝟏−𝐜𝐨𝐬 𝒕 𝒆𝒕−𝟏 , and 𝒕 𝒕 a. Multiplying the numerator and denominator by similar expressions. Example: sin 4𝑥 2 2 sin 4𝑥 𝐬𝐢𝐧 𝟒𝒙 ∙ = =2∙ 2𝑥 2 4𝑥 𝟒𝒙 30  𝐬𝐢𝐧 𝒕 Steps in Evaluating the Limits of Functions Involving the Expressions , 𝒕 𝟏−𝐜𝐨𝐬 𝒕 𝒆𝒕−𝟏 , and 𝒕 𝒕 b. Factoring either the numerator or denominator, or both. Examples: sin(𝑥+1) sin(𝑥+1) 1 𝐬𝐢𝐧(𝒙+𝟏) = = ∙ 𝑥 2 −1 (𝑥−1)(𝑥+1) 𝑥−1 𝒙+𝟏 2𝑥+4 2(𝑥+2) 𝒙+𝟐 = = 2 ∙ 𝑒 𝑥+2 −1 𝑒 𝑥+2 −1 𝒆𝒙+𝟐 −𝟏 31  𝐬𝐢𝐧 𝒕 Steps in Evaluating the Limits of Functions Involving the Expressions , 𝒕 𝟏−𝐜𝐨𝐬 𝒕 𝒆𝒕−𝟏 , and 𝒕 𝒕 c. Splitting the fraction as a sum, difference, or product of two or more fractions. Examples: sin 𝑥(1−cos 𝑥) 𝐬𝐢𝐧 𝒙 𝟏−𝐜𝐨𝐬 𝒙 = ∙ 𝑥2 𝒙 𝒙 𝑥 sin 𝑥+𝑒 −1 𝐬𝐢𝐧 𝒙 𝒙 𝒆 −𝟏 = + 𝑥 𝒙 𝒙 32  𝐬𝐢𝐧 𝒕 Steps in Evaluating the Limits of Functions Involving the Expressions , 𝒕 𝟏−𝐜𝐨𝐬 𝒕 𝒆𝒕−𝟏 , and 𝒕 𝒕 3. Apply the limit laws and the value of the special limits. 33 How will you obtain the 𝐬𝐢𝐧 𝒕 𝐬𝐢𝐧 𝟐𝒙 expression in ? 𝒕 𝐬𝐢𝐧 𝟖𝒙 34  Let’s Practice! 𝐬𝐢𝐧 𝟒𝒙 Evaluate 𝐥𝐢𝐦. 𝒙→𝟎 𝟑𝒙 35  Let’s Practice! 𝐬𝐢𝐧 𝟒𝒙 Evaluate 𝐥𝐢𝐦. 𝒙→𝟎 𝟑𝒙 𝐬𝐢𝐧 𝟒𝒙 𝟒 𝐥𝐢𝐦 = 𝒙→𝟎 𝟑𝒙 𝟑 36 Try It! 𝐬𝐢𝐧 𝟖𝒙 Evaluate 𝐥𝐢𝐦. 𝒙→𝟎 𝟓𝒙 37  Let’s Practice! 𝐬𝐢𝐧(𝒙+𝟏) Find the value of 𝐥𝐢𝐦. 𝒙→−𝟏 𝒙𝟐 −𝟏 38  Let’s Practice! 𝐬𝐢𝐧(𝒙+𝟏) Find the value of 𝐥𝐢𝐦. 𝒙→−𝟏 𝒙𝟐 −𝟏 𝐬𝐢𝐧(𝒙 + 𝟏) 𝟏 𝐥𝐢𝐦 =− 𝒙→−𝟏 𝒙𝟐 − 𝟏 𝟐 39 Try It! 𝐬𝐢𝐧(𝒙+𝟐) Find the value of 𝐥𝐢𝐦 𝟐. 𝒙→−𝟐 𝒙 −𝒙−𝟔 40  Let’s Practice! 𝒙−𝟏+𝐜𝐨𝐬 𝟐𝒙 Find the value of 𝐥𝐢𝐦. 𝒙→𝟎 𝟔𝒙 41  Let’s Practice! 𝒙−𝟏+𝐜𝐨𝐬 𝟐𝒙 Find the value of 𝐥𝐢𝐦. 𝒙→𝟎 𝟔𝒙 𝒙 − 𝟏 + 𝐜𝐨𝐬 𝟐𝒙 𝟏 𝐥𝐢𝐦 = 𝒙→𝟎 𝟔𝒙 𝟔 42 Try It! 𝟑𝒙−𝟏+𝐜𝐨𝐬 𝒙 Find the value of 𝐥𝐢𝐦. 𝒙→𝟎 𝟏𝟓𝒙 43  Let’s Practice! 𝟐 𝐬𝐢𝐧 𝟐𝒙+𝐜𝐨𝐬 𝟑𝒙−𝟏 What is the value of 𝐥𝐢𝐦 ? 𝒙→𝟎 𝟔𝒙 44  Let’s Practice! 𝟐 𝐬𝐢𝐧 𝟐𝒙+𝐜𝐨𝐬 𝟑𝒙−𝟏 What is the value of 𝐥𝐢𝐦 ? 𝒙→𝟎 𝟔𝒙 𝟐 𝐬𝐢𝐧 𝟐𝒙 + 𝐜𝐨𝐬 𝟑𝒙 − 𝟏 𝟐 𝐥𝐢𝐦 = 𝒙→𝟎 𝟔𝒙 𝟑 45 Try It! 𝟐 𝐬𝐢𝐧 𝟑𝒙−𝐜𝐨𝐬 𝟓𝒙+𝟏 What is the value of 𝐥𝐢𝐦 ? 𝒙→𝟎 𝟏𝟎𝒙 46  Let’s Practice! 𝒆𝟓𝒙 −𝟏 What is the value of 𝐥𝐢𝐦 𝟐𝒙 ? 𝒙→𝟎 𝒆 −𝟏 47  Let’s Practice! 𝒆𝟓𝒙 −𝟏 What is the value of 𝐥𝐢𝐦 𝟐𝒙 ? 𝒙→𝟎 𝒆 −𝟏 𝒆𝟓𝒙 − 𝟏 𝟓 𝐥𝐢𝐦 𝟐𝒙 = 𝒙→𝟎 𝒆 −𝟏 𝟐 48 Try It! 𝒆𝒙+𝟐 −𝟏 What is the value of 𝐥𝐢𝐦 𝟐 ? 𝒙→−𝟐 𝒆𝒙 −𝟒 −𝟏 49  Let’s Practice! 𝐬𝐢𝐧(𝒙−𝟑) What is the value of 𝐥𝐢𝐦 𝟐 ? 𝒙→𝟑 𝒆𝒙 −𝒙−𝟔 −𝟏 50  Let’s Practice! 𝐬𝐢𝐧(𝒙−𝟑) What is the value of 𝐥𝐢𝐦 𝟐 ? 𝒙→𝟑 𝒆𝒙 −𝒙−𝟔 −𝟏 𝐬𝐢𝐧(𝒙 − 𝟑) 𝟏 𝐥𝐢𝐦 𝟐 = 𝒙→𝟑 𝒆 𝒙 −𝒙−𝟔 −𝟏 𝟓 51 Try It! 𝐬𝐢𝐧(𝒙+𝟓) What is the value of 𝐥𝐢𝐦 𝟐 ? 𝒙→−𝟓 𝒆𝒙 +𝟗𝒙+𝟐𝟎 −𝟏 52  Check Your Understanding Use the provided table of values below to estimate the given special limits. left side of 𝟎 right side of 𝟎 𝑥 𝑓(𝑥) 𝑥 𝑓(𝑥) −0.5 0.5 1.2974425414 −0.1 0.1 1.0517091808 −0.01 0.01 1.0050167084 −0.001 0.001 1.0005001667 −0.0001 0.0001 1.0000500017 53  Check Your Understanding sin 𝑥 1. lim 𝑥→0 4𝑥 1−cos 6𝑥 2. lim 𝑥→0 2𝑥 𝑒 9𝑥 −1 3. lim 𝑥→0 3𝑥 54  Check Your Understanding Evaluate the following limits. sin 12𝑥 1. lim 𝑥→0 3𝑥 𝑒 2𝑥+5 −1 2. lim5 10𝑥+25 𝑥→−2 1−cos(𝑥+2) 3. lim 𝑥→−2 2𝑥+4 𝑒 2𝑥+1 −1 4. lim1 4𝑥2 −1 −1 𝑥→−2 𝑒 55  Let’s Sum It Up! To evaluate the limit of functions involving the sin 𝑡 1−cos 𝑡 𝑒 𝑡 −1 expressions , , and by algebraic 𝑡 𝑡 𝑡 method, we follow these steps: 0 1. Check if the function is indeterminate of type 0 at 𝑥 = 𝑐. 56 Let’s Sum It Up! 2. Manipulate the function to obtain the sin 𝑡 1−cos 𝑡 𝑒 𝑡 −1 expressions , , and. 𝑡 𝑡 𝑡 3. Apply the limit laws and the values of the sin 𝑡 1−cos 𝑡 𝑒 𝑡 −1 special limits lim , lim , and lim. 𝑡→0 𝑡 𝑡→0 𝑡 𝑡→0 𝑡 57 Challenge Yourself What is the value of 𝐬𝐢𝐧 𝟐 𝒙𝟐 −𝟐𝒙−𝟑 +𝐬𝐢𝐧(𝒙−𝟑) 𝐥𝐢𝐦 ? 𝒙→𝟑 (𝒙𝟐 −𝟗)(𝒙+𝟏) 58 Bibliography Edwards, C.H., and David E. Penney. Calculus: Early Transcendentals. 7th ed. Upper Saddle River, New Jersey: Pearson/Prentice Hall, 2008. Larson, Ron H., and Bruce H. Edwards. Essential Calculus: Early Transcendental Functions. Boston: Houghton Mifflin, 2008. Leithold, Louis. The Calculus 7. New York: HarperCollins College Publ., 1997. Smith, Robert T., and Roland B. Milton. Calculus. New York: McGraw Hill, 2012. Tan, Soo T. Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach. Australia: Brooks/Cole Cengage Learning, 2012. 59